Mathematical modeling and forecasting of seasonal characteristics of tourist flow

The article describes the flow of tourists to the Republic of Uzbekistan and the methods of analysis and forecasting based on econometric modeling of the development of the process based on its seasonal characteristics. Econometric modeling methods developed by foreign and local scientists were analyzed and divided into groups to analyze the process of changing the flow of tourists and predict the future number. Among them, the additive model in the group of time series reflecting the seasonality of tourist flow was found to meet the conditions. Based on the data obtained in the quarters for 2014-2018, the values of the trend (T), seasonal (S), and random (E) components of the time series were calculated step by step, and an additive model of the process was developed. Based on the developed model, the forecast values of tourist flow for the next quarter were determined, and the deviation from the actual value of the theoretical result was 20%, and the occurrence of this deviation was clarified. Forecasts of changes in the statistics of the tourism sector have been developed. The article describes the methods of analysis and forecasting of tourist flows and seasonality of the Republic of Uzbekistan based on econometric modeling.


Introduction
According to the World Tourism Organization (WTO), tourism accounts for 11 percent of world GDP, 10 percent of an investment, 11 percent of global consumer spending, and 5 percent of tax revenues. With this in mind, in recent years in the Republic of Uzbekistan, as in various countries worldwide, great attention is paid to the development of tourism [1].
The number of foreigners visiting the Republic of Uzbekistan is growing year by year. Over the past 16 years, the number of foreign visits to Uzbekistan has reached 5,346.2 thousand people, which is 15.5 times more than in 2002 [6].
Several scientific studies are currently being conducted to study the development of tourism and its role in the economy. Issues of rational use of the potential of the tourism sector in ensuring sustainable development of countries V.G. Gulyaev

Methods
Methods for assessing the impact of tourism on the sustainable development of the economy, the analysis of the system of models used to address the issues of forecasting the development of tourism in Uzbekistan was conducted by NN Safarova.
The research work analyzed above serves as a basis for scientific theoretical research in tourism, which are more regional approaches based on average annual data and statistical methods.
The main indicators of the development of modern tourism are the country's tourism infrastructure and the flow of tourists to the country. Therefore, in making optimal management decisions in the field, it is important to study the seasonal nature of the flow of tourists and their quantitative assessment and forecasting.
Our research focuses on assessing and predicting their seasonal characteristics based on econometric modeling of the flow of tourists to the country.

Results аnd Discussion
The issue of forecasting in tourism has always been one of the main issues in scientific research. To date, based on more than 150 models in tourism, many indicators, such as the number of tourists, tourism revenues, the role of tourism in the economy, can be predicted in the short and long term.
Given that the problem of forecasting has become so deep and expansive, the forecasting models used in research over the years by several researchers have been studied, analyzed, and the advantages and disadvantages of each model have been highlighted.
While Based on the significant findings from these studies, forecasting methods were divided into four groups ( Figure 1).
The time series models used in tourism forecasting are further divided into two groups; the first group includes analytical leveling, seasonality detection, sliding methods, and the second group, which includes complex methods, includes autoregression models and grinding models. The most preferred model among such models is the additive model, which is expressed in the form of a sum of trends, seasonality, cyclicality, and random events that reflect the characteristics of a time series. Autoregression models are formed using lags. Dozens of new models have been introduced into the scientific community based on the Box-Jenks methodology, i.e., the combination of autoregressive and sliding additive models. It is known that their seasonality characterizes the statistics in the field of tourism. According to the State Statistics Committee of the Republic of Uzbekistan, the change in the number of foreign tourists visiting the Republic of Uzbekistan in 2014-2018 by quarter confirms our opinion (Table 1). As can be seen from the data in the table, most of the tourist flow falls in the III quarter and a small part in the I quarter.
Processes with a seasonal character, expressed in time series, are studied by constructing additive or multiplicative models [3]. The additive model has a general view, in which each level (Y) of the time series consists of a set of trend (T), seasonal (S), and random (E) components.
The construction of the model consisted of several steps; first of all, the corresponding seasonal components' values were determined by aligning the given data series with the method of averaging (Table 2, row 3). The condition that the sum of the mean values of the detected seasonal components is zero must be met [2]. However, since the sum of the mean values of the seasonal components is different from zero, i.e., equal to -29633. 19   The condition is satisfied, which means that the values of the seasonal components are as follows: Using seasonal components (Table 3, column 3), the values of T + E = Y -S are found by subtracting the effect of the innocuous components from each level of a given time series (column 4 of Table 3). These values consist only of trend and random components.
To determine the T component of the model, we analyze the line (T + E) using a linear trend. We have a linear trend represented by the following equation [3]: By putting the values t = 1,2,…, 20 in this equation, the time values (levels) of T for each quarter are found (column 5 in Table 3). Then, by adding seasonal components for the quarters corresponding to the trend (T) levels, the theoretical values of the series according to the additive model are found (column 6 in Table 3).
The model process in which the time series constructed on the given (actual) values for tourist flow, the sum of the analytical textured trend and the theoretical values obtained in the model (columns 3 and columns 6), and the difference between them (table 3 column 7) is zero indicates an addictive reflection. The data in the graph and the results obtained show that the flow of tourists to our country tends to increase from year to year ( Figure 1).

Fig.1. Tourists visiting the Republic of Uzbekistan flow dynamics
According to the method of construction of the model, the model's error is calculated using the formula E = Y -(T + S) ( Table 4, column 7). accuracy.
This leads to the conclusion that the model can be used to solve the problem of forecasting.
We According to the website of the State Committee for Tourism Development of the Republic of Uzbekistan, in the first half of 2019, the flow of tourists to the Republic amounted to 3034824 people [5]. It can be seen that the difference in the model result is 20%. The reason for the sharp increase in the flow of tourists in the first half of 2019 is due to the ongoing reforms of the President of Uzbekistan in the field of tourism [2]. These figures show how well the model is structured. However, it should be noted that the forecast values can be calculated for a maximum of four years based on the given data, without changing the growth trend of the tourist flow.
To improve the forecast, forecasts for pessimistic (minimum), average and optimistic (maximum) cases for each year for 2020-2023 were developed (Table 4). The results of the table show that in 2023 compared to 2020, the number of tourists (visitors) in the Republic increased by 3092320 people (51%), revenues in the tourism sector to 274.4 billion US dollars (19%), the share of tourism in exports by 3.03%, the number of people employed in 27571 people ( 13%), and its share in GDP will increase by 0.22%. This shows the importance of tourism in the country's economy.

Conclusions
The analysis of the obtained data once again showed that the tourism industry plays an important role in a market economy and in the diversification of the economy and more intensive research. The results of our research show that the application of statistical methods and process modeling in the field of tourism has been carried out mainly over the years. Based on this, the system of models used in tourism was methodologically and practically analyzed and classified. It was found that the additive model fully approximates the process of modeling the seasonal flow of tourists in the tourism industry, and based on the website of the State Committee for Tourism Development of the Republic of Uzbekistan, developed an econometric model of tourist flow in the country. As a result, it was shown that the flow of tourists has an upward trend. At the same time, the indicators of the tourism sector are projected for 2020-2023, and it was found that they have a growing trend.